= Ising chain with alternating triangular decorations
{c}
{title2=$g_M=3^M+1$}
Take $2M$ base <Ising spins> on a ring and add one apex <Ising spin> over every other base bond. Give all base and apex bonds the antiferromagnetic <energy> $Jss'$ with $J>0$. There are $3M$ sites, and adjacent triangles are connected by a single undecorated bridge rather than sharing a corner. With $K=J/(k_BT)$, summing the apex spins yields alternating <spin-chain transfer matrices>
$$
A_{st}=2e^{-Kst}\cosh[K(s+t)],\qquad B_{st}=e^{-Kst},\qquad Z=\operatorname{Tr}(AB)^M.
$$
Their common symmetric <eigenvector> gives the leading cell <eigenvalue> $3+e^{-4K}+e^{-2K}+3e^{2K}$. A minimum-energy triangle has one frustrated bond and <energy> $-J$; its connecting bridge can have <energy> $-J$. Requiring the bridge to be antiparallel leaves the counting <transfer matrix> $\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)$ on successive first base spins. Its <eigenvalues> $3,1$ give <ground-state degeneracy> $g_M=3^M+1$ and <ground-state energy> $-2JM$. The <residual entropy> per site is $k_B\ln3/3$.
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